A Computational Investigation of a Turbulent Flow Over a Backward Facing Step with OpenFOAM

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1 206 9th International Conference on Developments in esystems Engineering A Computational Investigation of a Turbulent Flow Over a Backward Facing Step with OpenFOAM Hayder Al-Jelawy, Stefan Kaczmarczyk 2, Dhirgham AlKhafaji 3, Seyed Mirhadizadeh 4, Rob Lewis 5, Matt Cross 6, 2, 4 Faculty of Arts, Science and Technology, Engineering and Technology, The University of Northampton, UK. 3 College of Engineering, The University of Babylon, Iraq. 5, 6 TotalSim Ltd. UK. 2 3 Abstract-A computational fluid dynamics study of threedimensional turbulent flow over a backward facing step is presented. An available experimental study is investigated computationally using an open source tool. The wall static pressure distribution, the skin friction distribution and the reattachment length have been calculated and compared with the experimental data. Two different mathematical models were implemented using the OpenFOAM computational fluid dynamics (CFD) package. The long term goals for this research are to investigate and actively control the wake dynamics behind the step which will be useful to study the wake characteristics behind different types of bodies. Keywords OpenFOAM; Backwards Facing Step; CFD I. INTRODUCTION The backwards facing step (BFS) flow is a common and simple case to be investigated the flow behaviour. It has been a test case for numerical methods and turbulence models. The BFS case has some fundamental problems that could be transferred into much complicated cases [], [2]. Due to the wake dynamics of the BFS case, it is considered to be an optimal separated flow geometry. The separation location is a single fixed point where the flow separation occurs. This case shows all important simplifications with respect to more complex cases. There have been several experiments conducted over a backward-facing step geometry. The flow and geometry parameters of the case such as (aspect ratio, expansion ratio, free stream, Reynolds number, turbulence intensity as well as the boundary layer thickness at the separation point) were noted in the review paper by Eaton & Johnston [3]. Many experimentalists have studied comprehensively each of these parameters and their independent impacts [3], [4], [5]. Owing to the parametric differences between each experiments the reattachment length at step wake differs accordingly. In this paper, the incompressible flow and its characteristics measured experimentally by Driver & Seegmiller [5] has been studied computationally using OpenFOAM software [6]. Two theoretical models have been implemented (the two-equation k ω SST model and oneequation Spalart-Allmaras turbulence model) to compare the experimental results with the computer simulation results. II. THE MODEL DESIGN The model geometry (see Figure ) has an expansion ratio equal to 0.89 (it is a ratio of the inlet duct height to the height of the exit duct). The outlet boundary conditions take place downstream from the step which is 30h, where h denotes the step height. The experimental configuration also had an aspect ratio equal to 2 (it is a ratio of the duct width to the step height). This expansion ratio helps to reduce the freestream pressure gradient due to unexpected expansion at the separation location while the selected aspect ratio minimizes the three dimensional effects in the separated region. The test configuration is identified as shown in Table I. TABLE I. Component Inlet duct Length Height rectangular inlet duct Wide The step height THE MODEL GEOMETRY DIMENSIONS Dimension m 0.6 cm 5.24 cm.27 cm The experiments were performed at flow conditions listed in Table II. TABLE II. THE FLOW CONDITIONS Parameter Value Freestream velocity 44.2 m s Mach number 0.28 Reynolds number (Re) based on the momentu 5000 thickness at a position 4h upstream of the step The wall boundary layer thickness.9 cm /7 $ IEEE DOI 0.09/DeSE

2 Fig.. Mean flow structures of the turbulent backward facing step flow Fig. 2. Three-dimensional geometry of the case Fig. 3. Structured mesh of the backward facing step 302

3 Figure shows the mean flow structure that has been identified on the cross-sectional symmetry plane. Le et al [7] stated the fact that the structures shown in Figure do not present in the instantaneous flow and they generally illustrate the mean flow. The BFS mean flow characteristics are defined in Figure. The dividing streamline from the step tip to the reattachment location X s is defining the shear layer. The line between the reattachment point Y s and the re-separation point X s is called the primary-secondary interface line. This line separates the primary and secondary eddies. Figure 2 shows the three dimensional BFS geometry which was created by the Block mesh utility from OpenFOAM. There was no wall functions utilized in this study, so, the mesh at the walls was fine enough to capture the flow quantities. The structured mesh has been used and refined in the essential regions near the walls and at the step. Figure 3 shows the mesh resolution from four different 3-D views of the computational domain. The stagnation line u = 0 characterised the primary eddy. The secondary eddy is located between the two points defined by the coordinates X s and Y s, respectively. These points measure the interaction between the primary and secondary eddies. recognize the boundary layer and stop the switching between DES to LES mode inside it. These functions are presented below: F = tanh (arg 4 F ) (2) arg 4 F = min {min [max ( k, 500v ), β ωd d 2 ω 4σ ω2 k d 2 max( σ ω2 k ω ω xj xj,0 0 ) ]} (4) The turbulence kinematic viscosity coefficient v t is defined as: v t = a k max (a ω, S t F 2 ) S t is the strain rate (the vorticity magnitude) ( s) is shown below: S t = ( v i + v 2 j ) x j x j F 2 : is the second blending function which is defined as follows: (5) (6) F 2 = tanh (arg 4 F2 ) (7) arg 4 F2 = min [max ( 2 k, 500v β ωd d 2 ω ), 00] (8) The location of each eddy are defined by the stagnation points S p (x, y) and S s (x, y) as shown in Figure. According to Adams & Johnston [8], the velocity profile as well as the boundary layer thickness at the top δ t and the bottom δ b of the inlet duct characterise the inlet initial condition. III. MENTER S SHEAR STRESS TRANSPORT MODEL According to Menter [9], to close the Navier-Stokes equations by turbulent model, k ω model will be formulated into two equations governing the turbulence kinetic energy and specific dissipation rate as follows: (ρk) t + (ρku j) u = τ i x ij β ρωk + [(μ + σ j x j x k μ t ) k ] () j x j (ρω) + (ρωu j) = γ u τ i t x j v ij βρω 2 + 2( F t x ) σ ω2 k ω + j ω x j x j [(μ + σ x ωμ t ) ω ] (2) x j The transition between k ω model, applied near the wall, to the k ε model, applied in the freestream, occurs by the blending functions F and F 2. These functions help to σ k, σ ω, β, β and γ: are the closure coefficients in the turbulence rate equation (model constants); ρ: mean mass density; k: Turbulence kinetic energy; τ ij : Specific Reynolds stress/shear stress tensor; ω: Specific turbulent dissipation rate (turbulent frequency); ε: The turbulent dissipation (the rate at which velocity fluctuations dissipate); d: Length scale (the distance from the field point to the nearest wall). The various coefficients appearing in Menter s model according to Schaefer et al. [0] are determined as k ω model coefficients and k ε model coefficients as shown below: σ k = , σ ω = 0.5, β = 0.075, β = 0.09, γ = σ k2 = , σ ω2 = 0.5, β 2 = 0.075, β 2 = 0.09, γ 2 = Also, a = 0.3, c = 0.0, κ = 0.4 a : Bradshaw constant; β, γ: Coefficients in Wilcox k ω closure [0]; κ: Von Karman constant; σ k : Prandtl number in k-equation diffusion term; σ ω : Prandtl number in ω-equation diffusion term. IV. SPALART-ALLMARAS ONE EQUATION MODEL In 992, Spalart and Allmaras [2] developed a turbulent model for different turbulent flows, specifically aerodynamic flows which called Spalart-Almaras (SA) turbulence model. Basically, the transport equation of the eddy viscosity was 303

4 developed. The definition of the proposed model is shown in equation (9): DF = F Dt t + (u )F = Diffusion + Production Destruction (9) The three terms at the right hand side of equation (9) should be defined carefully in order to build the full turbulent flow model. The turbulence kinetic energy is not calculated in this model. The final form of the governing equation of SA model is shown below: v t v v v + u + v + w = c x y z b ( f t2 )S v [c w f w c b f κ t2] ( v d 2 )2 + [ ((v + v ) v v v ) + c σ x j x b2 ] (0) j x i x i The turbulent eddy viscosity is calculated as follows: v = ρf v μ t () f v = x3 x 3 +c3 ; x = ; The following equations will provide v v v additional definitions: Ω: The vorticity magnitude. v S = Ω + f κ 2 d 2 v2 (2) Ω = 2W ij W ij (3) d: The distance from nearest wall to the field point. Also, f v2 = f w = g [ +c 6 w3 6 g 6 6 ] +c w3 x (4) +xf v (5) V. THE SIMULATION SETUP 5 m2 The fluid kinematic viscosity is s. The boundary conditions for the simulation are shown in Table III. The turbulent intensity was 2%. Steady state calculations have been done with the same time step which is sec. The final 3504 time steps were analysed. TABLE III. THE BOUNDARY CONDITIONS Parameter Inlet Outlet Velocity 44.2 m s Zero Gradient Pressure Zero Gradient 0.0 Pa k (turbulent kinematic energy) m2 Zero Gradient s 2 ω (specific dissipation rate) 9.76 s Zero Gradient From mathematical point of view, setting up the boundary condition as a zero gradient value means it is Neumann boundary conditions for imposing the values of the solution derivative. The numerical simulation techniques were applied to solve the mathematical models. The solution was obtained by using the SimpleFoam tool, a Steady state Reynolds Averaged Navier Stokes (RANS) solver. It solves the incompressible turbulent flow based on SIMPLE algorithm [3]. VI. A. Reattachment length RESULTS AND DISCUSSION The fluid entrained causes an adverse pressure gradient which will cause the reattachment of the shear layer at a location known as the reattachment point. One of the primary parameters of the wake characteristics of the BFS flow is the reattachment length which is presented in this section. Figure 4 shows the difference on the velocity profile for three different cases. g = r + c w2 (r 6 r) (6) r = min [ v S κ 2 d 2, 0] (7) f t2 = c t3 exp( c t4 x 2 ) (8) W ij = 2 ( u i x j u j x i ) (9) The following coefficients have been used in the calculations: c b = 0.355, c b2 = 0.622, σ = 2 3, c w2 = 0.3, c w3 = 2, κ = 0.4, c t3 =.2, c t4 = 0.5 c v = 7. c w = c b κ + + c b2 2 σ Fig. 4. Velocity profile at a location 4 step-heights upstream of the step 304

5 Fig. 5. Contour plots of mean velocities and pressure in the recirculation region The Y-axis represents the boundary layer thickness and the X-axis represents the velocity profile. It is clear that the momentum thickness is growing up with extending the inlet duct length based on the boundary layer concepts. Four inlet duct lengths have been considered in order to achieve the same Reynolds number at the same position as experiments (,.,.2,.4 m). The inlet duct length in the experiments was m (see Table I.) and Reynolds number was 5000 (see Table II.). In order to achieve the same boundary layer thickness to obtain the same Reynolds number at the same position, the inlet duct length should be increased. The lengths of the shear layer reattachment are shown in table IV. with respect to the mesh density and the inlet duct length. The steady state k ω SST turbulence model was chosen. It is clear that the inlet duct.2 m (Case 2) provides the closest results to the experimental data of Driver & Seegmiller [5]. TABLE IV. k ω SST MODEL Case No. of Cell Inlet du Momentum Reattachment length (m thickness (mm location (mm) Experiments Case 768, Case 2 547, Case 3 675, TABLE V. SPALART_ALLMARAS MODEL Case No. of Cel Inlet du Momentum Reattachment length (m thickness (mm location (mm) Experiment Case 4 768, Case 5 52, Case 6 547, Figure 5 shows contours of the mean velocity and pressure that occur in the recirculation region. The mean flow structure could be noticed clearly from the mean velocity contour. Case (2) from Table IV and case (5) from Table V were chosen for calculate the pressure distribution, skin-friction distribution. The comparison between the k ω SST model and Spalart-Allmaras model shows a better qualitative agreement between the Spalart-Allmaras model and the experimental results in terms of the reattachment location as shown in Table VI. Also, utilizing the one-equation Spalart-Allmaras model is cheaper in terms of the computational cost. TABLE VI. TWO DIFFERENT TURBULENCE MODELS Case Simulation time (s) Mesh density (cells) Reattachment Locati (mm) Case , Case , The same calculations were carried out for the Spalart- Allmaras model. Table V. illustrates that the inlet duct length in SA model could be. m (Case 5) to have the same experiment conditions before the step. 305

6 distribution for the step side wall and for the opposite wall as shown in Figures 6 and 7, respectively. C. Skin-friction distribution The skin friction coefficient is defined by: Where: C f = τ w : The local wall shear stress. τ w 2 ρ U 2 (2) Fig. 6. Wall static pressure distribution for the step side wall of two different models Fig. 8. Skin-friction distribution over the step side wall B. Wall static pressure distributions The static pressure coefficient was calculated by using the relationship: Fig. 7. Wall static pressure distribution for the opposite wall C p = P P 2 ρu 2 (20) P : The freestream pressure; P: The local evaluated pressure at a certain point; ρ: is the fluid density (the density of air at Kg sea level and 5 C is.225 m 3 ); U : The freestream velocity. It is interesting to observe that k ω SST turbulence model performs better than S-A model in terms of the static pressure In Figure 8, the skin-friction fluctuation has been investigated on the step side wall. It shows that the S-A model provides a good agreement with experiments in the separation region. However, when the boundary layer starts to grow up after the reattachment point, k ω SST model gives a closer values than S-A model. VII. CONCLUSION Turbulent flow over the backwards facing step has been simulated using an OpenFOAM open source software. The steadiness of the flow behind the step was investigated by two models. The steady state scenario was studied by using the two-equation k ω SST model and the one-equation Spalart- Allmaras turbulence model. This solution has been developed by utilizing the SimpleFoam codes. Incompressible Newtonian fluid has been studied. The aim of this study was to check and test the computational results and compare them with the results of experiments in order to assess the codes robustness and realism. In this study, the finite volume method was used 306

7 to simulate the flow inside the backwards facing step. The results obtained have shown a good agreement with the experiments. It is evident that, the OpenFOAM incompressible solvers are relevant to be applied in such cases and could be taken into more complex cases. Further analysis include more interesting features might be the subject of further research. REFERENCES [] Gürbüz, M. and Tezer-Sezgin, M., 206. MHD Stokes flow in lid-driven cavity and backward-facing step channel. European Journal of Computational Mechanics, pp.-23. [2] Wang, B. and Li, H., 206. POD analysis of flow over a backward-facing step forced by right-angle-shaped plasma actuator. SpringerPlus, 5(), pp.- 9. [3] Eaton, J.K. & Johnston, J.P., 98. A review of research on subsonic turbulent flow reattachment. AIAA journal, 9(9), pp [4] Armaly, B.F. et al., 983. Experimental and theoretical investigation of backward-facing step flow. Journal of Fluid Mechanics, 27, pp [5] Driver, D.M. & Seegmiller, H.L., 985. Backward-facing step with inclined opposite wall. experiments by driver and seegmiller. Available at: cases/case30&cas30_head.html&cas30_desc.html&cas30_meth.html&cas30 _data.html&cas30_refs.html&cas30_rsol.html&&&&&&unknown [Accessed December 26, 205]. [6] OpenFOAM, Foundation, (205). Features of OpenFOAM. [Online] Openfoam.org. Available at: [Accessed 7 April 206]. [7] Le, H., Moin, P. and Kim, J., 997. Direct numerical simulation of turbulent flow over a backward-facing step. Journal of fluid mechanics, 330(), pp [8] Adams, E.W. and Johnston, J.P., 988. Effects of the separating shear layer on the reattachment flow structure part 2: Reattachment length and wall shear stress. Experiments in Fluids, 6(7), pp [9] Menter, F.R., 994. Two-equation eddy-viscosity turbulence models for engineering applications. AIAA journal, 32(8), pp [0] Schaefer, J. et al., 205. Uncertainty Quantification of Turbulence Model Closure Coefficients for Transonic Wall-Bounded Flows. In 22nd AIAA Computational Fluid Dynamics Conference. p [] Wilcox, D.C., 998. Turbulence modeling for CFD, DCW industries La Canada, CA. [2] Spalart, P.R. & Allmaras, S.R., 992. A one equation turbulence model for aerodinamic flows. AIAA journal, 94. [3] Page, M., Beaudoin, M. and Giroux, A.M., 200. Steady-state capabilities for hydroturbines with OpenFOAM. In 25th IAHR Symposium on Hydraulic Machinery and Systems, IOP Conference Series: Earth and Environmental Science (Vol. 2, No., p ). IOP Publishing. 307

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